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Brenda is choosing a car insurance plan. Based on her driving history and traffic where she lives, Brenda estimates that there is a [tex]$20\%$[/tex] chance she will have a car collision this year. In each plan, the insurance will cover the full cost of the collision after the deductible is paid.

Which of the plans detailed in the table is most likely to save Brenda the most money, based on expected value?

\begin{tabular}{|l|c|c|c|c|}
\hline
Plan & Deductible & Collision & Comprehensive & Premium Total \\
\hline
A & \[tex]$300 & \$[/tex]500 & \[tex]$234 & \$[/tex]734 \\
\hline
B & \[tex]$500 & \$[/tex]430 & \[tex]$212 & \$[/tex]642 \\
\hline
C & \[tex]$1,000 & \$[/tex]340 & \[tex]$175 & \$[/tex]515 \\
\hline
D & \[tex]$2,500 & \$[/tex]278 & \[tex]$127 & \$[/tex]405 \\
\hline
\end{tabular}

A. Plan A

B. Plan B

C. Plan C

D. Plan D

Sagot :

GinnyAnsweridn
To determine which car insurance plan is most likely to save Brenda the most money based on expected value, we need to calculate the expected cost for each plan. Expected cost is determined by taking into account both the fixed premium and the potential cost of a collision, adjusted for the probability of the collision occurring.

Let's break down the costs for each plan:

1. Plan A:
- Deductible: \[tex]$300 - Collision: \$[/tex]500
- Comprehensive: \[tex]$234 - Premium Total: \$[/tex]734

Expected Cost = Premium + (Probability of Collision * (Deductible + Collision Cost))
Expected Cost = \[tex]$734 + \(0.2 \times (\$[/tex]300 + \[tex]$500)\) Expected Cost = \$[/tex]734 + [tex]\(0.2 \times \$800\)[/tex]
Expected Cost = \[tex]$734 + \$[/tex]160
Expected Cost = \[tex]$894 2. Plan B: - Deductible: \$[/tex]500
- Collision: \[tex]$430 - Comprehensive: \$[/tex]212
- Premium Total: \[tex]$642 Expected Cost = Premium + (Probability of Collision * (Deductible + Collision Cost)) Expected Cost = \$[/tex]642 + [tex]\(0.2 \times (\$500 + \$430)\)[/tex]
Expected Cost = \[tex]$642 + \(0.2 \times \$[/tex]930\)
Expected Cost = \[tex]$642 + \$[/tex]186
Expected Cost = \[tex]$828 3. Plan C: - Deductible: \$[/tex]1,000
- Collision: \[tex]$340 - Comprehensive: \$[/tex]175
- Premium Total: \[tex]$515 Expected Cost = Premium + (Probability of Collision * (Deductible + Collision Cost)) Expected Cost = \$[/tex]515 + [tex]\(0.2 \times (\$1,000 + \$340)\)[/tex]
Expected Cost = \[tex]$515 + \(0.2 \times \$[/tex]1,340\)
Expected Cost = \[tex]$515 + \$[/tex]268
Expected Cost = \[tex]$783 4. Plan D: - Deductible: \$[/tex]2,500
- Collision: \[tex]$278 - Comprehensive: \$[/tex]127
- Premium Total: \[tex]$405 Expected Cost = Premium + (Probability of Collision * (Deductible + Collision Cost)) Expected Cost = \$[/tex]405 + [tex]\(0.2 \times (\$2,500 + \$278)\)[/tex]
Expected Cost = \[tex]$405 + \(0.2 \times \$[/tex]2,778\)
Expected Cost = \[tex]$405 + \$[/tex]555.60
Expected Cost = \[tex]$960.60 Now, we compare the expected costs: - Plan A: \$[/tex]894
- Plan B: \[tex]$828 - Plan C: \$[/tex]783
- Plan D: \[tex]$960.60 The lowest expected cost is \$[/tex]783 for Plan C.

Therefore, the plan that is most likely to save Brenda the most money, based on expected value, is:

C. plan C
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